From what I gather from Maeder’s list of publications, he’s an astrophysicist who recently had the idea to revolutionize cosmology by introducing a modification of general relativity. The paper which now makes headlines studies observational consequences of a model he introduced in January and claim to explain away the need for dark matter and dark energy. Both papers contain a lot of fits to data but no consistent theory. Since the man is known in the astrophysics community, however, the papers got published in ApJ, one of the best journals in the field.

For those of you who merely want to know whether you should pay attention to this new variant of modified gravity, the answer is no. The author does not have a consistent theory. The math is wrong.

For those of you who understand the math and want to know what the problem is, here we go.

Maeder introduces a conformal prefactor in front of the metric. You can always do that as an ansatz to solve the equations, so there is nothing modified about this, but also nothing wrong. He then looks at empty de Sitter space, which is conformally flat, and extracts the prefactor from there.

He then uses the same ansatz for the Friedmann Robertson Walker metric (eq 27, 28 in the first paper). Just looking at these equations you see immediately that they are underdetermined if the conformal factor (λ) is a degree of freedom. That’s because the conformal factor can usually be fixed by a gauge condition and be chosen to be constant. That of course would just give back standard cosmology and Maeder doesn’t want that. So he instead assumes that this factor has the same form as in de Sitter space.

Since he doesn’t have a dynamical equation for the extra field, my best guess is that this effectively amounts to choosing a weird time coordinate in standard cosmology. If you don’t want to interpret it as a gauge, then an equation is missing. Either way the claims which follow are wrong.
I can’t tell which is the case because the equations themselves just appear from nowhere. Neither of the papers contain a Lagrangian, so it remains unclear what is a degree of freedom and what isn’t. (The model is also of course not scale invariant, so somewhat of a misnomer.)

Maeder later also uses the same de Sitter prefactor for galactic solutions, which makes even less sense. You shouldn’t be surprised that he can fit some observations when you put in the scale of the cosmological constant to galactic models, because we have known this link since the 1980s. If there is something new to learn here, it didn’t become clear to me what.

Maeder’s papers have a remarkable number of observational fits and pretty plots, which I guess is why they got published. He clearly knows his stuff. He also clearly doesn’t know a lot about modifying general relativity. But I do, so let me tell you it’s hard. It’s really hard. There are a thousand ways to screw yourself over with it, and Maeder just discovered the one thousand and first one.

My take is that Maeder's \kappa vector is actually a gradient, so it can be gauge-fixed and taken out of the theory altogether. As you noted, the theory then just reverts to standard cosmology. Weyl tried this in 1918 with \kappa a true vector field, and that didn't work, either.

You can add vector fields to general relativity, it's not a big problem. You can also change the covariant derivative by allowing for vector non-metricity, or you can define a new effective metric with an additional term from a vector-field and couple with that. These modifications are all consistent; they can and have all been done, in particular the latter is quite popular for dark matter alternatives. But none of them is what Maeder is doing.

Here's my impression of the beginning of Maeda's paper "Scale invariant cosmology I: the vacuum and the cosmological constant", arXiv:1605.06315.

He likes position-dependent scale transformations of the metric, but Einstein's equations are not preserved by these transformations. So when he takes a solution of Einstein's equation and does a position-dependent scale transformation to it, the new rescaled metric is a solution of Einstein's equation only if this scale transformation obeys an equation. He works out this equation. In doing so, he allows the cosmological constant for the new solution to be different from the original cosmological constant.

Then he considers this equation in a special case: he starts with the simplest solution of the vacuum Einstein's equation with cosmological constant zero, namely Minkowski space, and assumes the scale transformation depends only on the t coordinate. He makes some handwaving argument about why this is interesting.

It's easy to solve the equation in this case. The result is that the new rescaled metric can be a solution of the vacuum Einstein equations with a positive cosmological constant if it describes deSitter spacetime - an exponentially expanding universe.

We can summarize most of this as follows: if you take Minkowski spacetime and apply a position-dependent scale transformation to the metric, you can get anti-deSitter spacetime. This is not news: it's discussed in any good book on general relativity.

Yes. But that's just saying that de Sitter space is conformally flat. More troubling is that he lumps the same prefactor from de Sitter in front of a general FRW later. Now, if you haven't fixed the scale factor already, this amounts to using a different ansatz for FRW, so you effectively reparameterize the time-coordinate. You can still solve the equations, all right, but that's not modified gravity, that's a modified coordinate system. Something else that you could do is actually add a dynamical equation for the factor, but I can't find anything in the paper that hints at that.

If a modified coordinate system enables an "explanation" of, say, galactic rotation curves, physically does that hint that time as measured by atomic phenomenon and time as measured by cosmology might not be quite the same thing?

The debate concerns two main points: 1) The expression of the scale factor and 2) The alleged underdetermined equations.

-1.) The conformal or scale factor must be a function of time only, for reason of homogeneity and isotropy. A value going like the inverse of the time allows one to apply the Minkowski metric in the scale invariant empty space, moreover it is a consistent solution of the general scale invariant equations for the empty space.

-2.) The cosmological constant expresses the density of the empty space. Thus,its relation to the scale factor is taken that of the empty space. This allows one to eliminate the cosmological constant from the equations, which nicely simplify. With the scale factor determined under point 1), the system of cosmological equations is fully defined.

This gives cosmological solutions with small, but interesting differences with respect to those of the LambdaCDM model. Applications to galaxies also cast doubt on dark matter,for which dozens of interpretation have been proposed.

I do not further enter a blog debate rich in personal attacks. The leading blogger will unavoidably have the last word.

There are many theories, the observations will choose one, not necessarily mine, but I will respect the exploratory efforts of¨everybody.

I don't know what you think is a "personal attack" here. I'm simply asking what are the equations you are solving. I have now asked this question like half a dozen times in different variants but not gotten an answer.

Your point 1) is correct if your matter is also homogeneous, which clearly isn't the case for galaxies.

As to 2), I don't know what you mean. My best guess is that you say if you put a factor 1/t^2 in front of an arbitrary metric, it creates an asymptotic de Sitter solution which is wrong. Even so, it doesn't answer my question. You can of course just take a solution to the usual equations, then lump a factor in front of it, and reinsert it into the equations. But then either the usual equations won't be fulfilled, or you have to recalculate the other metric coefficients (in case the ansatz still works), which doesn't change the solution.

So let me iterate this once again: What equations are you solving. Can you please just name them? Best,

I guess by personal attacks he means sentences like the following: "Maeder’s papers have a remarkable number of ... pretty plots, which I guess is why they got published".I have to say that I also dislike such sneering and patronizing comments. On top of being disrespectful, they are just a distraction and take away much of the fun from reading your otherwise very interesting and informative blog.

One has a general scale invariant field equation. There is some constant in it, the cosmological constant which represents the energy density of the empty space. I use the boundary condition of a vanishing matter density to express this constant in the differential equations. The metrics are the appropriate ones. Moreover, the constant is considered as independent on the metrics and systems considered.

Who is "one"? Do you mean you? which equation (singular?) are you referring to? What do you mean by scale invariant? Certainly you do not mean scale invariant in the usual way if you have matter added?

"I use the boundary condition of a vanishing matter density to express this constant in the differential equations."

That's clearly inconsistent with homogeneity.

"Moreover, the constant is considered as independent on the metrics and systems considered."

I'm not asking about the constant, I'm asking about the function, \lambda.

Didn't you tell Ryan that you agree the system is underdetermined? So what's this about now?

Might it be that trying to modify, or reformulate, general relativity to accommodate dark matter and dark energy be like trying to - squeeze blood out of a stone - to use an old metaphor? Perhaps an end run, to use a football analogy, from a wholly different angle is what's needed to break the deadlock. Such an attempt was made by Mordecai Milgrom with a version of his MOND theory, that posited a reduction in the inertial mass of bodies subject to a very low acceleration threshold. It seems to be quite successful in explaining the rotation curves of galaxies, but apparently has trouble explaining gravitational lensing from foreground dark matter.

S.H: Who is "one"? Do you mean you? which equation (singular?) are you referring to? What do you mean by scale invariant? Certainly you do not mean scale invariant in the usual way if you have matter added?A.M.: I mean everybody doing the work properly, including me. This is equation (7)of my paper, first derived by Canuto et al.(1977) from expressions obtainedby Eddington (1923) and Dirac(1973).

S.H.: That's clearly inconsistent with homogeneity.A.M.: In the limit of zero density, space is rather homogeneous!

S.H.: I'm not asking about the constant, I'm asking about the function, \lambda.A.M.: The constant is expressed in terms of \lambda and its derivative.

S.H.:Didn't you tell Ryan that you agree the system is underdetermined? So what's this about now?A.M.: What I told to Ryan is not a secret. As a consequence of what I said in my previous answer, the system is no longer underdetermined and it produces interesting new solutions.

Isn't it positive to try to remove some of the abundance of concepts from astrophysics? It is easy to expand a model to account for something, e.g. dark matter and dark energy. It is much harder to reduce your logic to explain something, but this is the correct way, if you use axioms in your logic. Though I haven't read his article, I humbly disagree with your viewpoint, Sabine.

Why believe in dark matter and dark energy? They are like the fifth element in ancient greek elemental theory. Shadowy and very much unproven/indirect concepts

Letting the density go to zero *everywhere* is not a boundary condition, that's a limiting case. And that case is again de-Sitter space. I already told you above that it's not consistent to use a fixed background in two of the equations, but not in the others, that's just nonsense.

Equation (7), to say it once again, is underdetermined if that's the only equation you use. That can't possibly be so hard to see. If you don't have kappa, the equations are fully determined. If you add a degree of freedom, you need an additional equation, or you can fix it by a gauge to zero. All you have to do is count!

Btw, I am getting pretty tired of this. You clearly aren't thinking about what I tell you. Why don't you go and ask anyone else who has worked on modifications of gravity, I am sure they will confirm what I tell you. This is pretty basic stuff.

Equation (7) as such is clearly underdetermined. But it contains the cosmologicalconstant \Lambda, the properties of which are independent of the coordinate system or space you consider, even de Sitter space. Thus, the expression based on the properties of the empty space, can (and must) be used to express \Lambda, which precisely represents the energy density of the empty space. Then, the system is no longer underdetermined. I know who is not listening to the other one.

So let me ask you once again, what is the additional equation that you use? Note that this additional equation should NOT have a fixed background, it should contain g_\mu\nu and \kappa (and \Lambda if you want, but that's tangential).

BTW, my previous comment is probably wrong for not all has being said and done! There is at least one case that seeminlgy has not being treated well as yet. It is the peculiar case first pointed out by Bahram Mashhoon where a collective of accelerated observers may induce a kind of nonlocality (of a "memory" type.) The peculiarity is that his aparently correct argumentation does not confront the axiomatic basis nor does it posits a need for modifying GR at all. All that this says if I am interpreting it correctly is that even basic GR is even more difficult mathematically then we usually like to believe!

Equation (20) is the equation relating the g_\mu\nu, \kappa and \Lamba in the empty space. There is a misunderstanding \Lambda is not tangential at all, but it is essential in the scale invariant context. It is \Lambda which prevents the equation of General Relativity to be scale invariant.

If this comment section were closed, you wouldn't be able to submit a comment... You also might want to look up what an argument from authority is. If I say I'm tired of repeating myself, that's not an argument from authority.

Having said that, it is pretty basic stuff to make sure that you have enough equations to determine all free functions. It is also pretty basic stuff that an equation which you derive in a special case from another equation is not an additional independent equation. It is also pretty basic stuff that FRW is not conformally flat, or that the standard model is not conformally invariant.

Equation (20) is what you get from Equation (7) in empty space (see section header) for a specific ansatz. This is neither an independent equation, nor can you use it for any ansatz that is not of the form which you assume there.

I say \Lambda is tangential for what I am saying because it's just some free parameter in the equation. I don't care about the parameter. I am referring to the number of equations.

Sitting back watching expert’s debate, having a good understanding human nature and some science history, I’m inclined to agree with David about attempting to modify general relativity. Einstein needed to rewrite Newtonian mechanics to incorporate his ideas that better fit evolving physics knowledge. Just as likely, general relativity will need to be as drastically rewritten, preserving dynamics it proved such as, an inseparable space-time, the malleability of space-time, the consistency in the speed of light for observers.

General Relativity preserved the functional dynamics of Newton, which can still be used as good approximation in many common applications today. However it took centuries to rewrite and refine them. Unfortunately it might take centuries more to rewrite General Relativity. It’s an often unacknowledged byproduct of human nature’s effect on scientific progress.

Right! I was trying to say, very politely, that in this paper Maeder not really doing anything interesting. Perhaps I was being too polite. I'll restate my comment a bit less politely, and also less technically, so nonexperts can understand it a bit better.

In his paper "Scale invariant cosmology I: the vacuum and the cosmological constant", Maeder is not really proposing a new theory of physics, although he seems to think he is. He's really just noting that you can take one solution of general relativity, do something to it, and get another. The first solution describes flat spacetime; the second one describes an expanding universe.

First of all, this not news. It's in every good textbook, and I learned about it in my college course on general relativity.

Second of all, it's a mathematical fact... but it's not a theory, or at least not a new theory. It's a mathematical fact about an old theory: general relativity, Einstein's theory of gravity. He's noticing some familiar stuff we can do with this theory. But he seems to think he's proposing a new theory.

Reading this paper was enough to make me uninterested in studying his later papers, with their more complicated errors.

Once again, I tell you that the parameter \Lambda about which you don't careis essential. In General Relativity, it is a free parameter. However, in the scale invariant theory, \Lambda is no longer a free parameter, as you are saying. Its specific meaning, as the density of the empty space (which I repeatedly mentioned), imposes a supplementary condition which is present AT ANY TIME. In other words, the cosmological constant \Lambda representsalways the energy density of the empty space and its relation to the scale factor, derived in the appropriate context, must apply.

Expressing this property gives a supplementary equation ALWAYS connecting \Lambda and some function of the scale factor and its derivatives. Thus, the number of equations is OK and the scale invariant cosmological equations are fully determined. As also nicely demonstrated by their consistent solutions.

In answer to John BaezThe scale invariant theory was originally developed in the context of the cotensorial calculus,first established by Weyl (1923), Eddington (1923), Dirac (1973) and Canuto et al. (1977). It is a theory, which incorporates one more invariance with respect to General Relativity. What I proposed is to use the above mentioned condition in this mathematical context, which leads to interesting solutions. It is amazing how many opposed reactions it provokes. By the way, the theory is generally not present in textbooks or college courses.

Of course \Lambda is relevant if you want to solve the equations. I am telling you merely that its presence is entirely irrelevant to the question whether your theory makes sense even on a basic level.

"Expressing this property gives a supplementary equation ALWAYS connecting \Lambda and some function of the scale factor and its derivatives."

Where is the equation? Please don't tell me once again it's equation (20), we have already debunked that. Equation (20) follows from the field equations. It's not an additional equation. It is also only fulfilled when the metric has certain properties which are not fulfilled in general.

Quoting you: ..." It is also only fulfilled when the metric has certain properties which are not fulfilled in general."

Thus, your statement implies that the properties of the empty space should notbe expressed with a metric which is that for the empty space, and as a consequence, it should change according to the physical context considered. This is incorrect. The properties of the empty space represented by a constant have to be always the same, as given by equations (24), whatever the physical context. I firmly maintain this point, which is our central divergence.

Sabine Hossenfelder wrote:"What do you mean by scale invariant? Certainly you do not mean scale invariant in the usual way if you have matter added?"

I also still don't understand what he means with "scale invariant theory/model".Maybe a GR model with those scale factor lambda conditions.

Scale invariant tensors seem to be tensors, which don't acquire a factor upon multipying the metric with a factor.

John Baez wrote:"It's easy to solve the equation in this case. The result is that the new rescaled metric can be a solution of the vacuum Einstein equations with a positive cosmological constant if it describes deSitter spacetime - an exponentially expanding universe."

I now briefly looked at

https://arxiv.org/pdf/1605.06315.pdf

What I understood by this brief look is that he says that a vacuum GR with a time varying cosmological constant of 3/c^2t^2 has the Minkowski metric as a solution. That is he says that if he uses a timevarying factor lambda^2 in front of the Minkowski metric, (which is a solution to the non-cosmological constant vaccuum GR) then for a general vaccum GR solution with this lambda scaled metric the cosmological constant can be written as a term which depends on time times a corresponding time-varying lambda - if the non-cosmological constant vaccuum equations hold as well. Than he says OK so let lambda=1 (i.e.Minkowski metric), so the cosmological "constant" is by this set to 3/c^2t^2. So apriori this seems a different "type" of GR, that is I don't know if this is still divergence free. At other instances (as I already said I briefly glanced at https://arxiv.org/pdf/1701.03964.pdf) he seems to replace the cosmological constant in a similar manner, just there he keeps the lambda and its specific timedependency apparently allows him to derive a different "Hubble value" (if I understand correctly a nonstatic Hubble "constant").

What about letting the "nature" decide ?Experts of the field have concluded... Mader's proposal is "not new theory theory" or is "mathematically" wrong. For J.B. et al. it's useless to read "his later papers, with their more complicated errors".

Ok ! you're probably right (aren't your experts ?). J.B. if you take the time to read the last paper, you will see the "pretty plots" (that are not only here to screw the referee and make the paper published). Could we also consider the predictions of Maeder's model ? What's the probability to reproduce by chance a set of observational constraints ? Couldn't we search for data in which this model fail ? I'm always more incline to believe the theorists that are able to reproduce the observational data. It makes the difference between theology and science.... Amen !

I am not interpreting your equations at all, I am merely saying you don't have enough.

Now you say it's eq (24), but this follows from (20) which follows from (7). It's still not an additional equation to fix your additional degrees of freedom.

You can, of course, as I already said above, use (24) as a gauge condition on the metric. Nothing wrong with that, but then you merely have general relativity in strange coordinates. (Provided that you actually solve the field equations correctly, which you may not have done, but that's another issue.)

So what is it? Will you finally let us know what the missing equation is? We're all holding our breath.

I know it's hard to swallow for some people, but mathematical consistency has turned out a very strong predictor of a theory's success. It adds to this that if you have a theory that no one except for its inventor can use to make predictions, that doesn't live up to the scientific standard.

If you put a factor 1/t^2 in front of the Minkowski metric, you get empty de Sitter space, ie a space with a cosmological constant. This means empty de Sitter space is conformally flat, which is hardly news. I believe he just wants to put the same factor in front of the FRW metric, but for FRW this merely amounts to redefining the time-coordinate, and this ansatz will in general just not work at all.

See, the usual ansatz for FRW is

dt^2 - a(t)^2 (dr^2 + d\Omega^2)

(for k=0 - doesn't matter for the sake of the argument). This will give the usual Friedmann equations.

But you can also make the ansatz

l(t) dt^2 - l(t)*a(t)^2 (dr^2 + d\Omega^2)

which will give you a different equation for a(t) which also depends on l(t). You can then go and fix l(t) to whatever you want, but you'll still describe the same space-time (unless you change the matter content of course).

Ie, such a "modification" has no physical relevance. You can of course make the additional function a dynamical field and give it a kinetic term, but then you should have an additional equation for this field (or at least a constraint).

S.H.: You can, of course, as I already said above, use (24) as a gauge condition on the metric. Nothing wrong with that, but then you merely have general relativity in strange coordinates

A.M.: Equation (24), which is the supplementary equation, is not general relativity in strange coordinates (!), this is just the relation between the energy density of the empty space and the scale factor in the appropriate coordinates.

I think the whole problem is that you do not want to accept that THE PROPERTIES OF THE SCALE INVARIANT EMPTY SPACE LEADING TO EQ. (24) HAVE TO BE ALWAYS THE SAME, INDEPENDENTLY OF SPACE AND TIME. These properties constitute the supplementary equation closing the system. It is true, as you pointed it, that this equation comes from the field equation. But, it represents a supplementary condition to be accounted for, in orderto express the properties of the cosmological constant.

TO THE PARTICIPANTS,

I will leave this debate, at least for the moment. I have emphasized above the basic property, that in my opinion closes the system.One may like it or not. The fact is that it is the simplest and most natural one.Future observations and comparisons will tell us whether this gauge is theone chosen by Nature or not. On my side, I am going on with this exploration.Finally, it is nice to see so many people interested in the properties of ...... the empty space. Bye !

I did not say that eq (24) is general relativity in strange coordinates. I don't even know what sense this statement makes. What I said is that if eq (24) is a gauge, then your "modified" Friedmann equations (27,28) are the usual ones (or they are just wrong, which is also an option, frankly I didn't check).

I don't know what you think all caps will help you. I have asked a simple question, and it's evident that you cannot answer it.

Sabine Hossenfelder wrote: "If you put a factor 1/t^2 in front of the Minkowski metric, you get empty de Sitter space, ie a space with a cosmological constant."

OK. Aha. Maybe. I only know the description from Hawking and Ellis, there they go to Minkowski R5 and there de Sitter space is the hyperboloid. It is also that the last time I listened to a class in GR was 30 years ago and it was a more experimental class. Anyways how I understood Maeder in https://arxiv.org/pdf/1605.06315.pdf he constructs a spacetime with a time-variable cosmological constant with the Minkowski metric as a solution because he sets lambda=1.

In https://arxiv.org/pdf/1701.03964.pdf for the case of the Friedmann-Robertson-Walker metric he apparently plugs in the above condition (from https://arxiv.org/pdf/1605.06315.pdf) between cosmological constant and lambda and assumes that lambda behaves as for this above model that is in particular that lambda_dot/lambda=-1/t, which gives a different Hubble value. Again he assumes a time-variable cosmological "constant" and not a constant constant. Thats how I understand it, but I really only briefly looked at this and I am not really sure what to think of it. What I find interesting is that usually it seems you have rather strong conditions -at least in 3 dimensions- if you want to keep the connection torsion free and want that the symmetric tracefree part of the Ricci tensor vanishes. But I don't know how the connection looks like here or the Ricci tensor.

I think I get it. The same equation is used twice to close the system but one instance of it is restricted by the scale invariance of empty space. So, they are not really twice the same equation, more like one for space with matter which is not scale invariant, and the other for empty space which is scale invariant. It is this additional hypothesis which enables this. This looks like say, using Newtonian equations within GR, which is clearly just a limit and not always true, but the scale invariance of empty space is always there, even if the space in consideration is not exactly empty. Empty space is embedded in non-empty space, so the more empty the space is, the more impact this scale invariance of empty space has. In short, empty space and non-empty space are treated separately and thus require their own specifications; they can come from the same equation, but both need to be in appropriate form.

I suspect the only kind of equation that SH or JB would consider as an acceptable one for a field is one that is derived from the extremisation of an action for this field ... it's kind of nowadays theoretical physics dogma

I'm sorry but i still don't understand why you absolutely exclude the possibility that some degrees of freedom could satisfy equations which would be derived in another way, or even directly demanded to satisfy new kind of symmetry principles (here apparently it's scale invariance) ... what's wrong with this ? of course demanding that lambda depends on time only would be inconsistent with the requirement that lambda be a totally dynamical field with equations derived as usual , but the author here just decided to stay away from this usual imperative rule. As far as the equations are generally covariant or coud easily be rewritten in a generally covariant way i don't see what's fundamentally wrong with this way of building theories.

Moreover i'm wondering exactly when did theorists start to block themselves with such unnecessary mental barriers such as : everything should follow from an action and everything in an action should be dynamical! In the 70s there was a lot of theorical attempts with fixed background non dynamical metrics : they called them prior geometry metrics and Clifford Will has reviewed some of them in his book.

You don't necessarily need a Lagrangian to get a consistent set of equations, but it's a sufficient condition. If you don't have a Lagrangian you have to do some other check to make sure you have at least enough equations/constraints to fix your degrees of freedom. At the very least you should count them. And tell us what the degrees of freedom are! Having said that, personally I wouldn't waste my time on a modification of gravity that doesn't have a Lagrangian. It's one thing to believe that a fundamental theory doesn't have a Lagrangian. Actually quite plausible. It's another thing entirely to believe that the IR limit doesn't have a Lagrangian. Best,

I understand it's a dangerous game when you have equations which do not follow from a lagrangien. Here at first glance we have an extra dof which is lambda but there is also an extra equation which is just lambda(tau)~ 1/tau (equation 17). It does not follow from a lagrangien yet i'm ready to accept it because it directly follows from the new principles of this theory : homogeneity and isotropy of the metric lambda . eta_munu even when the sources are not homogeneous and isotropic , and scale invariance of the theory.

Now could you give us a feeling of what could go wrong with the dofs here because again at first sight there is just an extra dof and also an extra equation for it... ? do you suspect something wrong with the Bianchi identities in the modified GR equation ? can you be more specific ?

I have been very specific above. How much more specific do you need it? You can of course put such a factor in front of the FRW metric but that's merely a reparameterization of the metric. No problem with that, but also no new physics. That's just the same physics in a funny gauge. As I emphasized above, that's because this is not an independent equation. It follows from the field equations for a certain ansatz.

which will give you a different equation for a(t) which also depends on l(t). You can then go and fix l(t) to whatever you want, but you'll still describe the same space-time (unless you change the matter content of course)."

That's not at all the game he is playing. the solution with this new ansatz in the GR field equation remaining the same of course would of course give back the same solution, but the solution of the new ensatz with a new field equation (his modified GR equation) is not the same as the solution with the old ensatz (FRW) in the GR equation. Actually, as usual it's the equations that determine the solution and not the ensatz. In his case i do agree that the field equation alone is not enough : indeed there is the crucial external constraint that lambda(t) depends on time only!

So i do agree however that page 4 of the paper 1701.03964.pdf is a bit misleading as it does not make sufficiently clear that there is not only a field equation but also this external constraint on lambda (here homogeneity and isotropy of lambda whatever everything else)

What is less trivial is why this constraint alone is enough to completely determine lambda : my intuition is that in GR cosmology we have two equations to be satisfied by the scale factor but actually one of them is automatically satisfied if the source energy-momentum tensor is conserved (because of the Bianchi identities).Here because the einstein equation is modified it might be that there is really an additional independent cosmological equation which helps completely determining all dofs ... (?)

But it's the same equation! Yes, if you had a *different* modified equation, then you could do that (though I don't recommend it, but let's leave this aside for a moment). But it's not. It's the same equation! Look, to see what I mean just take lambda = constant and the rest will be the usual equations. And that's a perfectly fine and allowed solution. It's just that the ansatz isn't of the form that is postulated in the paper.

Look, forget that particular solution. Just ask the general question, without solving anything. Where are the additional equations?